Weighted Tur&amp;#xe1;n type inequality for rational functions with prescribed poles

Zhao, Dan; Zhou, Songping; Yu, Dansheng; Wang, Jianli

doi:10.7153/jmi-08-17

Cited by 2 publications

(2 citation statements)

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“…Suppose z ∈ K B , then |M − z| |1 + z|. Applying this, inequality (22) from Lemma 5, and estimate (30) we obtain…”

Section: Construction Of a Polynomial And A Partition Of The Set Kmentioning

confidence: 82%

“…Yu and Wei [21, Corollaries 1 and 2] obtained Turán type inequalities for doubling, and in case of q = ∞, for so-called A * weights. Subsequently, in [22] the results for doubling weights were extended to a somewhat larger class of weights, called "N-doubling weights". Some L q results were obtained for the disk and its perimeter with the (unweighted) arc length measure on it, but truly weighted versions were not derived.…”

Section: Introductionmentioning

confidence: 99%

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Turán type oscillation inequalities in $L^q$ norm on the boundary of convex domains

Glazyrina¹,

Révész²

2017

Mathematical Inequalities &Amp; Applications

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Abstract. Some 77 years ago P. Turán was the first to establish lower estimations of the ratio of the maximum norm of the derivatives of polynomials and the maximum norm of the polynomials themselves on the interval I := [−1,1] and on the unit disk D := {z ∈ C : |z| 1} under the normalization condition that the zeroes of the polynomial p all lie in the interval or in the disk, respectively. He proved that with n := deg p tending to infinity, the precise growth order of the minimal possible ratio of the derivative norm and the norm is √ n for I and n for D . J. Erőd continued the work of Turán and extended his results to several other domains. The growth of the minimal possible ratio of the ∞ -norm of the derivative and the polynomial itself was proved to be of order n for all compact convex domains a decade ago.Although Turán himself gave comments about the above oscillation question in L q norms, till recently results were known only for D and I . Here we prove that in L q norm the oscillation order is again n for a certain class of convex domains, including all smooth convex domains and also convex polygonal domains having no acute angles at their vertices.Mathematics subject classification (2010): 41A17, 30E10, 52A10.

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“…Suppose z ∈ K B , then |M − z| |1 + z|. Applying this, inequality (22) from Lemma 5, and estimate (30) we obtain…”

Section: Construction Of a Polynomial And A Partition Of The Set Kmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%